Benoit Mandelbrot, beginning in the 1960s, observed and recorded the properties of many natural phenomena, from turbulence in fluids to the stock market. Now we have the basic terms with which to describe fractals, we can examine some natural processes. If the fractal object is a surface, and it is displaced in a third dimension (i.e.: like a fractal landscape), then the fractal dimension is between 2.0 and 3.0. This gives us log 4 / log 3 = 1.26 as the fractal dimension of the curve. The above replacement rule replaces a section of perimeter with a section 4/3 times the length. We calculate it using the following formula:įractal Dimension = log(Length iteration n+1) / Log(Length iteration n) It is, if you like, a measure of roughness of the surface. In the case of a one-dimensional example such as the Koch Snowflake (where the fractal object is a perimeter rather than a surface), the fractal dimension is between 1.0 and 2.0. This is a measure of the increase in complexity at each iteration. To help us describe fractals, we use a measure called 'fractal dimension'.
#Terragen classic erosion full
In the case of the Koch Snowflake (the above shows just the top - the full figure does resemble a snowflake) we have a recursive geometric description of the set, with each new approximation to the fractal a little closer to the final, infinitely complex shape. If we continue this for about 5 iterations, we get the following shape: REPLACE ALL LINES AS FOLLOWS, AND REPEAT 2. If it were possible to continue to n = INF, we would either find |Z INF| to be infinite, in which case the initial point, C, lies outside the set, or to be finite, in which case C lies inside the set.Īnother sort of fractal is best described in geometric terms, using a rule of replacement. If we begin with Z 0 = 0 + 0i and C = x + y i, and we calculate successive terms, |Z| (the size of Z) either becomes larger and rushes towards infinity, or it stays close to zero, trapped in a so-called basin of attraction.
#Terragen classic erosion series
An example of such a series is that which describes the Mandelbrot Set: In most cases, this series has no analytic solution, so all we can hope to do is to converge on the correct solution by calculating successive terms in the series. In other words this means that, for example, the position of a each point on a fractal path is given by the sum of an infinite series. The key to fractals is that they have recursive definitions. We put a value for the radius in, we get 4 angles out in the range 0 -> 2p.įor a fractal object, this ceases to be the case. Such formulae are simple mappings - we put in a value of the angle in, we get a value of the radius out. Thus, an ellipse might be expressed as a function of radius and angle in polar coordinates. In conventional geometry, shapes are defined by simple formulae. Those who have a more complete understanding of this field will notice that there are some complexities that are glossed over. The purpose of this page is not so much to provide a mathematically complete and rigorous account of the techniques used for fractal landscape generation, but rather to provide an overview of these techniques and their application that is accessible to someone with high-school level maths. The Maths of Fractal Landscapes and Procedural Landscape Generation
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